Suppose a computer program needs to apply an affine transformation to a complex three-dimensional object made up of 7000 points. The transformation is composed of 7 matrices (call them $M_1$ through $M_{7}$), so for each point $(x, y, z)$ in the object, the following operation is performed. $\left[\begin{array}{c} ~ \\ ~ M_1 ~ \\ ~ \end{array}\right] ~ \left[\begin{array}{c} ~ \\ ~ M_2 ~ \\ ~ \end{array}\right] ~ \left[\begin{array}{c} ~ \\ ~ M_3 ~ \\ ~ \end{array}\right] ~ \cdots \left[\begin{array}{c} ~ \\ ~ M_{6} ~ \\ ~ \end{array}\right] ~ \left[\begin{array}{c} ~ \\ ~ M_{7} ~ \\ ~ \end{array}\right] ~ \left[\begin{array}{c} x \\ y \\ z \\ 1 \end{array}\right]$ Each multiplication of a matrix times a column vector involves 16 multiplications (of one number by another) and 12 additions, for a total of 28 arithmetic operations. Each multiplication of a matrix times another matrix involves 64 multiplications and 48 additions, for a total of 112 arithmetic operations. (These numbers are not made up or chosen randomly; they are facts about $4\times 4$ matrix multiplication.) The most inefficient way of applying the transformation to the 7000 points would be to begin on the left, multiplying $M_1$ by $M_2$, then that result by $M_3$, and so on along the list from left to right, and doing the same 7 multiplications again for each of the 7000 points. How many arithmetic operations would this require for transforming one point? Answer: How many would it require in total for transforming all 7000 points? Answer: A more efficient method would be to start on the right, first multiplying $M_{7}$ by the column vector, then multiplying $M_{6}$ by that result, and so on, proceeding along the list from right to left. How many arithmetic operations would this require for transforming one point? Answer: How many would it require in total for transforming all 7000 points? Answer: Using this method would therefore save what percentage of the time of the previous method? Answer: % The most efficient method would be to multiply the 7 matrices together and call that result $A$ (without yet multiplying $A$ by any column vector). After computing $A$ (just once of course), multiply $A$ by each of the 7000 points, each represented as a column vector. How many arithmetic operations would it take to compute $A$? Answer: How many arithmetic operations would it require to multiply $A$ by one point? Answer: How many would this method require in total? Answer: Using this method would therefore save what percentage of the time of the previous method? Answer: %